First-Principle Measurements: An Important Concept

Concepts in Natural Gas Measurement:

First-principle; Reducing Uncertainty

What is a direct first-principle measurement? Why is it important? What does it have to do with uncertainty? What is “compounding uncertainty” in indirect measurements?

Rather than spending time on definitions, let’s start by an example:

Imagine we have an object that is a square cube such as the picture below:

We measure each side to be 10 cm, but the ruler we are using has a 10% uncertainty. So we say each side is 10±1 cm. Now, we are asked to calculate the volume. Well that is an easy calculation. The volume of the cube is 10cmx10cmx10cm = 1000 cm3. But the uncertainty of this volume calculation is 30%! Why?

The minimum volume could be is 9x9x9 = 729 cm3. The maximum it could be is 11x11x11 = 1331 cm3. Notice that while we had only a 10% uncertainty in the measurement of each side, we are now up to about 30% uncertainty in calculation of the volume from those measurements!!! What happened?!!

The answer is that uncertainties compound in calculations[1]. In general, when you add two numbers, the uncertainty of the sum is the sum of the two individual uncertainties. When you multiply (or divide two numbers) you should add the uncertainties in the % form to get the total % uncertainty.

Is there a better way to measure the volume to reduce this uncertainty?!! Yes. It would be better to measure the volume directly, for example by immersing it in a graduated cylinder and looking at the rise in the water level (see picture below).

This is measuring the volume directly. We are only limited by the uncertainty of the graduated cylinder. There are no calculations involved. Notice that when we measure the volume directly, we have eliminated another source of uncertainty as well. What if the cube was not a perfect cube? That would have added another source of error when calculating. When measuring the volume directly, the shape is no longer important. It could be a cube, a pyramid or a sphere. It would not make a difference!!

The above is an example of direct first principle measurement. We were interested in the volume and we measured it as directly as possible; i.e. we measured the property we were interested in, rather than inferring it through other properties and calculations. When we measure a property directly and it is based on a core physical principle, then we are making a first-principle measurement.

How is this important in making measurements in general? How is it important when deciding what to measure and how to measure it?

It is typically true that direct measurements afford the possibility of minimizing uncertainties. So whenever we need to make measurements, we have to consider the core property that matters to our process, the most direct method for measuring it, the tools that are available, and the uncertainty of those tools.

As an important example of this principle for natural gas measurements, consider the case of moisture measurement. Let’s compare measuring moisture dew point vs measuring the moisture content using an aluminum (ceramic oxide).

The dew point of the gas only depends on moisture level and the pressure of the gas (there is a small dependence on gas composition as well). Dew point is measured by cooling an element, detecting condensation, and recording the temperature of condensation. The figure of merit is the temperature, and as long as the temperature sensor is reliable (and you know the condensation is water), then you have a measurement limited by the temperature sensor uncertainty (typically very low).

Now let’s consider moisture measurement using an aluminum oxide sensor. First, moisture has to travel through a diffusion barrier which supposedly limits diffusion to only water molecules (in reality this is not the case). So you have to assume a constant diffusion rate across this barrier. This means, that the barrier is also not partially covered by any contaminants (not a very good assumption for natural gas). Then, the water molecules that make it across the barrier will change the capacitance of a capacitor depending on how many molecules made it to the capacitor. Now the change in this capacitance is measured using some circuitry. Using calibration tables, one infers how much moisture there was in the gas. In order for this measurement to work, you have made the following assumptions:

1- The diffusion rate of water molecules across the barrier is constant.

2- No other molecule will impact the capacitance (interference).

3- The change of capacitance per water molecule does not drift over time!

4- The capacitance measurement circuitry has low uncertainty.

5- Factory calibrations of the sensor have low uncertainty.

These assumptions, in general, are rarely true for moisture measurements in natural gas. Furthermore, each of those assumptions and measurement have an uncertainty associated with them. For example, the flow of moisture molecules across the barrier has some uncertainty associated with it. So does the change in capacitance, the factory calibrations, etc. etc. All of these uncertainties will be compounded to produce an uncertainty that is much bigger than expected.

Comparing moisture dew point measurements with moisture measurements using an aluminum oxide is a good example to illustrate first-principle measurement and low uncertainty associated with them. Although the discussion was about aluminum oxide, the same considerations apply when using Tunable Diode Lasers (TDL) or Quartz Crystal Monitors (QCM). In later blogs we will address TDL and QCM in more detail.

EPILOGUE:

Many manufacturers of moisture measurement instruments (such as aluminum oxide, QCM, and TDL) erroneously call their instruments dew point sensors. THEY ARE NOT.

Dew point can only be measured using the chilled-mirror principle. These other instruments measure moisture content (with all the compounded uncertainty) and then calculate a dew point. Calling these instruments Dew Point analyzers, or Dew Meters, or any such variation on the phrase “Dew” is misleading.

So next time you are in the market for a dew point analyzer, ask the vendor if his instrument is truly a dew point analyzer, or if it is being called one for marketing purposes.

[1] A complete discussion of uncertainty requires a statistical approach.